Brazilian President Michel Temer was allegedly caught in a secret audio recording authorizing the head of meatpacking giant JBS SA to pay bribes to buy the silence of disgraced political leader Eduardo Cunha, who is serving a 15-year prison sentence for corruption.

The recording was delivered to the Supreme Court on Wednesday as part of evidence in a plea-bargain deal that JBS’s Joesley Batista is seeking in relation to the mammoth Lava Jato graft investigation, according to a report in the newspaper O Globo.

The paper’s website crashed in the hour after the story broke.

The news roiled Brazil, as a nation that thought itself inured after three years of political turmoil confronted a dramatic ratcheting up of the level of chaos and uncertainty.

Television channels cut from regular programing to air scenes from Brasilia of top political leaders rushing red-faced and panic-stricken through the hallways of Congress, on their way to meet with Mr. Temer.

The President’s office released a statement confirming he met with Mr. Batista in March but denying the allegations: “President Michel Temer never solicited payments to obtain the silence of former [Congress] deputy Eduardo Cunha. He never participated and never authorized any action with the objective of avoiding testimony or co-operation with the justice system by the former member.”

The Lava Jato investigation has involved a succession of stunning revelations but never anything like this: all previous plea bargains have related to the recounting of bribery and other illegal actions committed in the past. But Mr. Batista, and his brother Wesley, with whom he runs JBS, appear to have been working together with police to entrap political figures in criminal activity.

Mr. Batista allegedly told Lava Jato prosecutors that he has paid $2.2-million to Mr. Cunha just since last October. The tape reportedly records the President saying, “You need to keep this up, got it?”

Mr. Cunha, an evangelical preacher, was the Speaker of the lower house of Congress and the most powerful man in the country until his ignominious downfall. He spearheaded the impeachment of former president Dilma Rousseff last year. Mr. Temer recently told an interviewer that Mr. Cunha told him he would make sure she was ousted after she refused to shut down an investigation into his actions. In March, Mr. Cunha was sentenced to 15 years for corruption and money-laundering, for taking bribes to ensure the awarding of lucrative state contracts – and then funnelling the money to Swiss bank accounts.

As he was forced out of office, Mr. Cunha warned that he had damaging information on virtually all of his political colleagues – and the O Globo story suggests that Mr. Temer and others have been keen to ensure he didn’t share that information.

Also named in the JBS plea bargain is Aecio Neves, the head of the Brazil Social Democratic Party (PSDB), who narrowly lost the last presidential race to Ms. Rousseff and was considered a contender for president in the next election. He was allegedly recorded asking Mr. Batista for $870,000 on March 24 to pay for expenses related to Lava Jato; he is under investigation for a range of financial crimes.

The Batista brothers have been charged with “crimes against the national finance system,” including misuse of loans from a national development bank.

The Lava Jato investigation has revealed an entrenched and widespread symbiotic relationship between Brazil’s largest businesses, particularly construction firms such as Odebrecht SA and agribusiness heavyweights like JBS. For decades, politicians have accepted off-the-books campaign funding and personal bribes from these companies in exchange for awarding enormous state infrastructure and procurement contracts and writing friendly legislation. A giant swath of the political class, including a third of senators and nearly a third of cabinet ministers, are under investigation or indictment in connection with Lava Jato.

Mr. Temer, alone, has been safe – until now. He has been named in a number of other plea bargains, relating to bribes he allegedly authorized before he took over as President with Ms. Rousseff’s ouster last August. But Brazilian law says he cannot be prosecuted for those crimes while he holds the office of President. However, the incidents in the reported recordings would seem to lead Brazil into new legal territory: these alleged crimes were committed just weeks ago.

“The owners of JBS made a joint plea bargain [to Supreme Court Justice] Edson Fachin, with recordings that link the President Michel Temer to Eduardo Cunha – something that is an atomic bomb,” Chico Alencar, an opposition member of Congress, said in the lower house when the news broke in the late evening. “It just keeps coming, in this country, because we have a corrupt and degenerate political caste.”

The tapes are theoretically part of a sealed deposition, but Brazil’s political crisis over the past two years has played out in a series of leaks to media. If the tapes prove to show what O Globo columnist Lauro Jardim has reported in his explosive story, it raises serious questions about Mr. Temer’s ability to hold the office until his term ends in late 2018.

In other countries, a president might see no option but to resign over news such as this, but this is less likely in Brazil, where Mr. Temer will likely hunker down and take refuge in a blizzard of court processes and bureaucracy.

Within an hour of the story being published, a member from Congress from an opposition party filed an impeachment request. Opposition members of Congress were reported to be negotiating the wording of a proposed constitutional amendment to try to force Mr. Temer out.

Many Brazilians also read into the fact that it was O Globo, a media conglomerate whose wealthy owners were viewed as backers of the Rousseff impeachment and supporters of Mr. Temer, that broke this story – a sign that he has lost their protection, which could complicate his efforts to see out his term.